Integral formulas for recovering extremal measures for vector constrained energy problems.

*(English)*Zbl 1434.31003Summary: Extremal problems for vector potentials have wide applications in asymptotic analysis of Hermite-PadĂ© approximants of analytic functions. We consider equilibrium vector logarithmic potentials with constrains on measures. We study the dependance of the supports of the equilibrium measures on their masses. We obtain the integral formulas for recovering the extremal measure of given mass from the supports of the equilibrium measures of smaller masses.

##### MSC:

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

PDF
BibTeX
XML
Cite

\textit{M. A. Lapik}, Lobachevskii J. Math. 40, No. 9, 1355--1362 (2019; Zbl 1434.31003)

Full Text:
DOI

**OpenURL**

##### References:

[1] | E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality (AMS, Providence, 1991). · Zbl 0733.41001 |

[2] | B. Beckermann, V. A. Kalyagin, A. C. Matos, and F. Wielonsky, “Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses,” Construct. Approx. 37, 101-134 (2013). · Zbl 1261.31001 |

[3] | A. I. Aptekarev and V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Pade approximants,” Mat. Sb. 201 (2), 29-78 (2010). · Zbl 1188.42009 |

[4] | A. I. Aptekarev, V. A. Kalyagin, and E. Saff, “Higher-Order three-term recurrences and asymptotics of multiple orthogonal polynomials,” Construct. Approx. 30, 175-223 (2009). · Zbl 1189.41007 |

[5] | Aptekarev, A. I., The Mhaskar-Saff variational principle and location of the shocks of certain hyperbolic equations (2016), Providence · Zbl 1350.41012 |

[6] | A. I. Aptekarev, V. G. Lysov, and M. A. Lapik, “Direct and inverse problems for vector logarithmic potentials with external fields,” Anal. Math. Phys. (in press). https://doi.org/10.1007/s13324-019-00297-8 · Zbl 1428.41016 |

[7] | V. G. Lysov and D. N. Tulyakov, “On a vector potential-theory equilibrium problem with the Angelesco matrix,” Proc. Steklov Inst. Math. 298, 170-200 (2017). · Zbl 1388.30043 |

[8] | V. G. Lysov and D. N. Tulyakov, “On the supports of vector equilibrium measures in the Angelesco problem with nested intervals,” Proc. Steklov Inst. Math. 301, 180-196 (2018). · Zbl 1401.31008 |

[9] | A. A. Gonchar and E. A. Rakhmanov, “On the equilibrium problem for vector potentials,” Russ. Math. Surv. 40, 183-184 (1985). · Zbl 0594.31010 |

[10] | P. D. Dragnev and E. B. Saff, “Constrained energy problems with applications to orthogonal polynomials of a discrete variable,” J. Anal. Math. 72, 223-259 (1997). · Zbl 0898.31003 |

[11] | V. S. Buyarov and E. A. Rakhmanov, “Families of equilibrium measures in an external field on the real axis,” Sb. Math. 190, 791-802 (1999). · Zbl 0933.31002 |

[12] | A. B. J. Kuijlaars and E. A. Rakhmanov, “Zero distributions for discret orthogonal polynomials,” J. Comput. Appl. Math. 99, 255-274 (1998). · Zbl 0929.33010 |

[13] | M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field,” Sb. Math. 206, 211-224 (2015). · Zbl 1314.31004 |

[14] | E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Vol. 316 of Grundlehren Math. Wiss. (Springer, Berlin, 1997). · Zbl 0881.31001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.